18.783 Elliptic Curves Lecture 1 - MIT Mathematics

18.783 Elliptic Curves Lecture 1

Andrew Sutherland

February 8, 2017

What is an elliptic curve?

The

equation

x2 a2

+

y2 b2

=

1

defines

an

ellipse.

An ellipse, like all conic sections, is a curve of genus 0. It is not an elliptic curve. Elliptic curves have genus 1.

The area of this ellipse is ab. What is its circumference?

The circumference of an ellipse

Let y = f (x) = b 1- x2/a2. Then f (x) = -rx/ a2 - x2, where r = b/a < 1. Applying the arc length formula, the circumference is

a

a

4

1 + f (x)2 dx = 4

1 + r2x2/(a2 - x2) dx

0

0

With the substitution x = at this becomes

1 1 - e2t2

4a

0

1 - t2 dt,

where e = 1 - r2 is the eccentricity of the ellipse.

This is an elliptic integral. The integrand u(t) satisfies

u2(1 - t2) = 1 - e2t2.

This equation defines an elliptic curve.

An elliptic curve over the real numbers

With a suitable change of variables, every elliptic curve with real coefficients can be put in the standard form

y2 = x3 + Ax + B, for some constants A and B. Below is an example of such a curve.

y2 = x3 - 4x + 6 over R

An elliptic curve over a finite field

y2 = x3 - 4x + 6 over F197

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